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Calculus 2 - Integration by Trigonometric Substitution £9.16
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Exam (elaborations)

Calculus 2 - Integration by Trigonometric Substitution

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This document will allow university Math students to practice integration by trigonometric substitution. All the questions are similar in style and full worked solutions have been provided. This will allow you to understand how a certain question in this document is answered and then you will be ab...

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  • November 13, 2024
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Calculus 2 - Integration by Trigonometric Substitution

Case 1:

∫ 𝑎² − 𝑥² 𝑑𝑥

Use substitution 𝑥 = 𝑎𝑠𝑖𝑛𝜃

𝑑𝑥 = 𝑎𝑐𝑜𝑠𝜃𝑑𝜃




Case 2:


∫ 𝑥² − 𝑎² 𝑑𝑥

Use Substitution 𝑥 = 𝑎𝑠𝑒𝑐𝜃

𝑑𝑥 = 𝑎𝑠𝑒𝑐𝜃𝑡𝑎𝑛𝜃𝑑𝜃




Case 3:

∫ 𝑥² + 𝑎² 𝑑𝑥 OR ∫ 𝑎² + 𝑥² 𝑑𝑥

Use Substitution 𝑥 = 𝑎𝑡𝑎𝑛𝜃

𝑑𝑥 = 𝑎𝑠𝑒𝑐²𝜃𝑑𝜃

, 1) ∫ 49 − 𝑥² 𝑑𝑥

Case 1:

𝑎² = 49
𝑎 = 49
𝑎 = 7
𝑥 = 7𝑠𝑖𝑛𝜃
𝑑𝑥 = 7𝑐𝑜𝑠𝜃𝑑𝜃

Substitute 𝑥 = 7𝑠𝑖𝑛𝜃 and 𝑑𝑥 = 7𝑐𝑜𝑠𝜃𝑑𝜃 into the original integral

∫ 49 − (7𝑠𝑖𝑛𝜃)² 7𝑐𝑜𝑠𝜃 𝑑𝜃

This integral is now in trigonometric terms

∫ 49 − 49𝑠𝑖𝑛²𝜃 7𝑐𝑜𝑠𝜃 𝑑𝜃

∫ 49(1 − 𝑠𝑖𝑛²𝜃) 7𝑐𝑜𝑠𝜃 𝑑𝜃

∫ 49𝑐𝑜𝑠²𝜃 7𝑐𝑜𝑠𝜃 𝑑𝜃

∫7𝑐𝑜𝑠𝜃 7𝑐𝑜𝑠𝜃 𝑑𝜃

∫49𝑐𝑜𝑠²𝜃 𝑑𝜃

49∫𝑐𝑜𝑠²𝜃 𝑑𝜃

Power reduction formula for cos²𝜃:

1 1
𝑐𝑜𝑠²𝜃 = 2
+ 2
𝑐𝑜𝑠2𝜃

1 1
49∫ 2 + 2
𝑐𝑜𝑠2𝜃 𝑑𝜃

1 1
49 ( 2 𝜃 + 4
𝑠𝑖𝑛2𝜃)

49 49
2
𝜃 + 4
𝑠𝑖𝑛2𝜃

,The expression in the green box is the answer however, we need to convert this
answer to an expression in terms of 𝑥 since the original question was in terms of 𝑥.
We will add the + C at the very end in the final answer to avoid any confusion.

Sin Double Angle Formula: 𝑠𝑖𝑛2𝜃 = 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃

49 49
2
𝜃 + 4
(2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃)

49 49
2
𝜃 + 2
𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃


𝑥 = 7𝑠𝑖𝑛𝜃 → This is the first substitution we used at the beginning of the question.

𝑥 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑠𝑖𝑛𝜃 = 7
→ 𝑠𝑖𝑛𝜃 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒




𝑥




49 − 𝑥²



𝑐² = 𝑎² + 𝑏²
7² = 𝑥² + 𝑏²
49 = 𝑥² + 𝑏²
𝑏² = 49 − 𝑥²
𝑏 = 49 − 𝑥²

49 49
2
𝜃 + 4
𝑠𝑖𝑛2𝜃

49 49
= 2
𝜃 + 2
𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃

𝑥
𝑠𝑖𝑛𝜃 = 7


𝑥
𝜃 = 𝑠𝑖𝑛⁻¹( 7 )

, 49−𝑥²
From Triangle: 𝑐𝑜𝑠𝜃 = 7


49 49
2
𝜃 + 4
𝑠𝑖𝑛2𝜃

49 49
= 2
𝜃 + 2
𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃

49 49
Substitute 𝜃, 𝑐𝑜𝑠𝜃 and 𝑠𝑖𝑛𝜃 into 2
𝜃 + 2
𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃

49 𝑥 49 𝑥 49−𝑥²
2
𝑠𝑖𝑛⁻¹( 7 ) + 2
( 7 )( 7
)


Simplifying the expression above:

49 𝑥 1
2
𝑠𝑖𝑛⁻¹( 7 ) + 2
𝑥 49 − 𝑥²


FINAL ANSWER:
49 𝑥 1
2
𝑠𝑖𝑛⁻¹( 7 ) + 2
𝑥 49 − 𝑥² + 𝐶

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